Theoretical and Numerical Study of the Blow up in a Nonlinear Viscoelastic Problem with Variable-Exponent and Arbitrary Positive Energy

2021 ◽  
Author(s):  
Ala A. Talahmeh ◽  
Salim A. Messaoudi ◽  
Mohamed Alahyane
Keyword(s):  
Variable Exponent ◽  
Blow Up ◽  
Numerical Study ◽  
Positive Energy ◽  
Viscoelastic Problem ◽  
Nonlinear Viscoelastic

scholarly journals Blow-Up of Solutions for a Class Quasilinear Wave Equation with Nonlinearity Variable Exponents

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Zakia Tebba ◽  
Hakima Degaichia ◽  
Mohamed Abdalla ◽  
Bahri Belkacem Cherif ◽  
Ibrahim Mekawy
Keyword(s):  
Wave Equation ◽  
Finite Time ◽  
Variable Exponent ◽  
Blow Up ◽  
Initial Energy ◽  
Positive Energy ◽  
Variable Exponents ◽  
Quasilinear Wave Equation ◽  
New Class ◽  
Dispersion Term

This work deals with the blow-up of solutions for a new class of quasilinear wave equation with variable exponent nonlinearities. To clarify more, we prove in the presence of dispersion term − Δ u t t a finite-time blow-up result for the solutions with negative initial energy and also for certain solutions with positive energy. Our results are extension of the recent work (Appl Anal. 2017; 96(9): 1509-1515).

Well-posedness and blow-up properties for the generalized Hartree equation

2020 ◽  
Vol 17 (04) ◽  
pp. 727-763
Author(s):  
Anudeep Kumar Arora ◽  
Svetlana Roudenko
Keyword(s):  
Finite Time ◽  
Blow Up ◽  
Hartree Equation ◽  
Energy Threshold ◽  
Local Theory ◽  
Positive Energy ◽  
Small Data ◽  
Well Posedness ◽  
Time Behavior ◽  
Mass Energy

We study the generalized Hartree equation, which is a nonlinear Schrödinger-type equation with a nonlocal potential [Formula: see text]. We establish the local well-posedness at the nonconserved critical regularity [Formula: see text] for [Formula: see text], which also includes the energy-supercritical regime [Formula: see text] (thus, complementing the work in [A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, Michigan Math J., forthcoming], where we obtained the [Formula: see text] well-posedness in the intercritical regime together with classification of solutions under the mass–energy threshold). We next extend the local theory to global: for small data we obtain global in time existence and for initial data with positive energy and certain size of variance we show the finite time blow-up (blow-up criterion). In the intercritical setting the criterion produces blow-up solutions with the initial values above the mass–energy threshold. We conclude with examples showing currently known thresholds for global vs. finite time behavior.

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